Evans, et al., [9] defined an equivalence relation ∼ on the set of vertices of a simple graph G by taking a ∼ b if and only if their open neighborhoods are the same. They introduced a new graph Gred = G/∼, reduction graph of G, as follows. The vertices are V (Gred) = {[a] : a ∈ V (G)}, and two equivalence classes [a] and [b] are adjacent if and only if a and b are adjacent in G. Recently, Anderson and LaGrange [4] defined some equivalence relations on the set of vertices of the zero-divisor graph of a commutative ring, one of which yields the reduction graph of the zerodivisor graph. In this paper, we state some basic graph theoretic properties of Gred and study the relations between some properties of graph G and its subgraph, G_red, such as the chromatic number, clique number, girth and diameter. Moreover, we study the reduction graph of some algebraic graphs, such as the comaximal graph, zero-divisor graph and Cayley graph of a commutative ring. Among other results, we show that, for every commutative ring R, Γ_2(R)red ≃ Γ1_(Zn 2 ), where Γ_1(Zn 2 ) is the zero-divisor graph of the Boolean ring Zn 2 , Γ2(R) is the comaximal graph of R and n = |Max(R)|.
